Multimode josephson parametric converter: coupling josephson ring modulator to metamaterial

ABSTRACT

A technique relates to a microwave device. The microwave device includes a Josephson ring modulator, a first multimode resonator connected to the Josephson ring modulator, where the first multimode resonator is made of a first left-handed transmission line, and a second multimode resonator connected to the Josephson ring modulator, where the second multimode resonator is made of a second left-handed transmission line.

DOMESTIC PRIORITY

This application is a continuation of U.S. patent application Ser. No.14/871,477, filed Sep. 30, 2015, the disclosure of which is incorporatedby reference herein in its entirety.

BACKGROUND

The present invention relates to quantum information processing in themicrowave domain using superconducting circuits, and more specifically,to a multimode Josephson parametric converter.

Recent progress in solid-state quantum information processing hasstimulated the search for amplifiers and frequency converters withquantum-limited performance in the microwave domain. Depending on thegain applied to the quadratures of a single spatial and temporal mode ofthe electromagnetic field, linear amplifiers can be classified into twocategories (phase sensitive and phase preserving) with fundamentallydifferent noise properties. Phase-sensitive amplifiers squeeze the inputnoise and signal in one quadrature of the microwave field at the expenseof inflating the noise and signal in the other quadrature without addingnoise of their own to the processed signal, but are useful only in casesin which the quantum information is encoded in one quadrature of themicrowave field. A phase-preserving amplifier on the other handamplifies both quadratures of the input noise and signal at the expenseof adding at least a noise equivalent to a half input photon at thesignal frequency. Such an amplifier would be useful in many quantumapplications, including qubit readout. One successful realization of anon-degenerate-intrinsically phase-preserving-superconducting parametricamplifier is based on a Josephson ring modulator, which consists of fourJosephson junctions in a Wheatstone bridge configuration. The devicesymmetry enhances the purity of the amplification process, i.e.,eliminates or minimizes certain undesired nonlinear processes, and alsosimplifies both its operation and its analysis.

SUMMARY

According to one embodiment, a microwave apparatus is provided. Themicrowave apparatus includes a Josephson ring modulator, a firstmultimode resonator connected to the Josephson ring modulator, where thefirst multimode resonator is made of a first left-handed transmissionline, and a second multimode resonator connected to the Josephson ringmodulator, where the second multimode resonator is made of a secondleft-handed transmission line.

According to one embodiment, a method of configuring a microwaveapparatus is provided. The method includes connecting a first multimoderesonator to a Josephson ring modulator, where the first multimoderesonator is made of a first left-handed transmission line, andconnecting a second multimode resonator to the Josephson ring modulator,where the second multimode resonator is made of a second left-handedtransmission line.

According to one embodiment, a microwave apparatus is provided. Themicrowave apparatus includes a Josephson ring modulator in a Wheatstonebridge configuration, where the Josephson ring modulator include a firstpair of nodes opposite one another and a second pair of nodes oppositeone another, and a first multimode resonator connected to the Josephsonring modulator, where the first multimode resonator is a firstleft-handed transmission line. The first multimode resonator isconnected to the first pair of nodes, and the first multimode resonatorincludes first unit cells. Also, the microwave apparatus includes asecond multimode resonator connected to the Josephson ring modulator,where the second multimode resonator is a second left-handedtransmission line. The second multimode resonator is connected to thesecond pair of nodes, and the second multimode resonator includes secondunit cells. A ground plane is connected to inductors respectively in thefirst unit cells and the second unit cells, and capacitors connect toanother end of the inductors respectively in the first unit cells andthe second unit cells.

According to one embodiment, a microwave apparatus is provided. Themicrowave apparatus includes a Josephson ring modulator in a Wheatstonebridge configuration, where the Josephson ring modulator includes afirst pair of nodes opposite one another and a second pair of nodesopposite one another, and a first multimode resonator connected to theJosephson ring modulator, where the first multimode resonator is a firstleft-handed transmission line. A lumped-element side of the firstmultimode resonator is connected to one node in the first pair of nodesand a conducting plane is connected to another node in the first pair ofnodes. The first multimode resonator includes first unit cells. Also,the microwave apparatus includes a second multimode resonator connectedto the Josephson ring modulator, where the second multimode resonator isa second left-handed transmission line. A lumped-element side of thesecond multimode resonator is connected to one node in the second pairof nodes, and a conducting plane is connected to another node in thesecond pair of nodes. The second multimode resonator includes secondunit cells.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a high-level schematic of a quantum microwave device accordingto an embodiment.

FIG. 2 is a circuit representation of a semi-infinite losslessleft-handed transmission line utilized in the multimode microwaveresonators of a multimode Josephson parametric converter according to anembodiment.

FIG. 3 is a schematic of the multimode Josephson parametric converteraccording to an embodiment.

FIG. 4 is a coplanar waveguide implementation of the multimode Josephsonparametric converter according to an embodiment.

FIG. 5 is a semi-coplanar stripline implementation of the multimodeJosephson parametric converter according to an embodiment.

FIG. 6 is a method of configuring a microwave apparatus according to anembodiment.

DETAILED DESCRIPTION

Embodiments disclose a quantum device based on the Josephson ringmodulator suitable for quantum information processing. The quantumdevice includes a Josephson ring modulator coupled to multimoderesonators implemented using metamaterial/left-handed transmissionlines, thereby forming a multimode Josephson parametric converter.

FIG. 1 is a high-level schematic of a quantum microwave device 100according to an embodiment. The quantum device 100 includes a multimodeJosephson ring modulator (JRM) 105 which is a nonlinear dispersiveelement based on Josephson tunnel junctions 102A, 102B, 102C, and 102Dthat can perform three-wave mixing of microwave signals at the quantumlimit. The JRM 105 consists of four nominally identical Josephsonjunctions 102A-102D arranged in a Wheatstone bridge configuration. Inorder to construct a non-degenerate parametric device that is themultimode Josephson parametric converter (JPC) 130, which is capable ofamplifying and/or mixing microwave signals at the quantum limit, the JRM105 is incorporated into two multimode microwave resonators at a radiofrequency (RF) current anti-node of the multiple of their eigenmodes.

One of the multimode microwave resonators is multimode resonator_a 115Aand the other is multimode resonator_b 115B. The multimode resonator_a115A is a left-handed transmission line with N unit cells, and themultimode resonator_b 115B is a left-handed transmission line with Munit cells as discussed further below. A coupling capacitor 110Aconnects the multimode resonator_a 115A to port_a 120A while thecoupling capacitor 110B connects the multimode resonator_b 115B toport_b 120B. The multimode JPC 130 includes both the multimoderesonator_a 115A and multimode resonator_b 115B, along with the JRM 105.

The performances (namely power gain G, dynamical bandwidth γ, andmaximum input power P_(max)) of the multimode JPC 130 are stronglydependent on the critical current I₀ of the Josephson junctions102A-102D of the JRM 105, the specific realization of theelectromagnetic environment (i.e., the microwave multimode resonator_a115A and microwave multimode resonator_b 115B), the coupling between theJRM 105 and the multimode resonators 115A and 115B, and the couplingbetween the multimode resonators to the feedlines.

The port_a 120A and/or port_b 120B may be microwave coaxial lines orwaveguides. Although not shown, other devices connected to the quantumdevice 100 may include hybrids, attenuators, circulators, isolators,lowpass microwave filters, bandpass microwave filters, infrared filters,and qubit-cavity systems.

FIG. 2 is a circuit of a semi-infinite lossless left-handed transmissionline which may be utilized in the construction of the multimodemicrowave resonator_a 115A and the multimode microwave resonator_b 115Baccording to an embodiment. The unit cell, e.g., unit cell 205A formicrowave multimode resonator_a 115A and unit cell 205B for microwavemultimode resonator_b 115B, includes a capacitor C₁ connected toinductor L_(l) where “l” represent left-handed transmission line. Theother end of the inductor L_(l) is connected to ground. The unit cell205A, 205B is connected to another unit cell, which is connected toanother unit cell, and so forth. The unit cell 205A is repeated N amountof times for the multimode resonator_a 115A, and the unit cell 205B isrepeated M amount of times for the multimode resonator_b 115B, as shownfurther below.

The dispersion relation of a left-handed transmission line reads

${\omega_{l}\left( k_{l} \right)} = \frac{1}{2\sqrt{L_{l}C_{l}}{\sin \left( \frac{k_{l}\Delta \; x}{2} \right)}}$

where Δx is the size of the unit cell, and k_(l) is the wave vector.

The phase and group velocity of the left-handed transmission line haveopposite orientation

${\frac{\partial{\omega_{l}(k)}}{\partial k} < 0},$

where k is k_(l). One consequence of this relation is that inleft-handedtransmission lines low-frequencies correspond to short wavelengths. Incontrast, in right-handed transmission lines where the dispersionrelation increases with the wave vector, low-frequencies correspond tolong wavelengths.

The characteristic impedance of the left-handed transmission line is

$Z_{l} = {\sqrt{\frac{L_{l}}{C_{l}}}.}$

Low-frequency bound of the left-handed transmission line is

$\omega_{IR} = {\frac{1}{2\sqrt{L_{l}C_{l}}}.}$

FIG. 3 is a schematic of the multimode Josephson parametric converter130 according to an embodiment. In FIG. 3, a 180° hybrid coupler 305Amay be connected to port_a 120A and a 180° hybrid coupler 305B may beconnected to port_b 120B.

A 180° hybrid is a 4-port microwave device which is reciprocal, matched,and ideally lossless. The 180° hybrid splits an input signal into twoequal amplitude outputs. When fed from its sum port (Σ) the 180° hybridprovides two equal-amplitude in-phase output signals, whereas when fedfrom its difference port (Δ), it provides two equal-amplitude 180°out-of-phase output signals.

One scenario assumes that there is a Signal (S) tone that lies withinthe bandwidth of one of the resonance modes of the multimode microwaveresonator_a which strongly couples to the JRM and is input through the γport of the 180° hybrid 305A, and the 50 ohm (Ω) termination isconnected to the Σ port of the 180° hybrid 305A. It also assumes thatthere is an Idler (I) tone that lies within the bandwidth of one of theresonance modes of the multimode microwave resonator_b which stronglycouples to the JRM and is input through the Δ port of the 180° hybrid305B and a pump (P) tone input into the Σ port of the 180° hybrid 305B.Note that multiple pump tones at different frequencies may be utilizedin order to feed the device.

The two main operation modes of the device are amplification mode (withphoton gain) in which the applied pump frequency f_(P) satisfies therelation

f _(P) =f _(I) +f _(S),

where f_(S) and f_(I) are the frequency of the Signal (S) and the Idler(I) tones respectively, and unitary frequency conversion mode (withoutphoton gain) in which the applied pump frequency f_(P) satisfies therelation

f _(P) =|f _(I) −f _(S)|.

Different implementations of the quantum device with the multimodeJosephson parametric converter 130 are discussed herein according toembodiments.

In contrast, to dual-differential-mode (standard nondegenerate)state-of-the-art Josephson parametric converters made of right-handedtransmission lines, e.g., microstrip resonators, where the JRM isstrongly couples to the two fundamental resonance modes of the twophysical resonators of the device within the frequency band of interest,e.g. 5-15 GHz, the two multimode resonators of the multimode JPC 130,realized using metamaterial/left-handed transmission lines inembodiments (i.e., multimode resonator_a 115A and multimode resonator_b115B), can be designed and engineered such that the JRM 105 stronglycouples to multiple differential modes within the frequency band ofinterest. That is, each multimode resonator_a 115A and multimoderesonator_b 115B has multiple resonance modes within the frequency bandof interest, e.g. 5-15 GHz, many of which strongly couple to the JRM105, as opposed to the state-of-the-art JPCs having (only) twofundamental differential resonance modes for its resonators within theband of interest which strongly couple to the JRM.

Multimode means that the multimode resonator_a 115A has multipleresonance modes and that the multimode resonator_b 115B has multipleresonance modes within a certain frequency band of interest, e.g. 5-15GHz. This means that multimode resonator_a 115A is configured toresonate at multiple resonance frequencies from a first resonancefrequency through a last resonance frequency within a certain frequencyband of interest, which may include hundreds of resonance frequencies.Similarly, the multimode resonator_b 115B is configured to resonate atmultiple resonance frequencies from a first resonance frequency througha last resonance frequency within a certain frequency band of interest,e.g. 5-15 GHz, which may include hundreds of resonance frequencies.

One notable property of the left-handed transmission lines/resonators(respectively implemented as multimode resonator_a 115A and multimoderesonator_b 115B) is that they have a large density of modes (i.e.,density of resonance modes) close to their low-frequency bound ω_(IR)making them multimode resonators in the frequency band of interest. Forquantum measurements in superconducting devices, the band of interest isthe microwave band of approximately 5-15 gigahertz (GHz) (commonly usedfor qubit readout and measurement). The multimode resonator_a 115A andmultimode resonator_b 115B each can have a high density of resonancemodes (i.e., harmonics or resonance frequencies) between approximately5-15 GHz, which is beneficial for quantum measurements. In contrast, aright-handed transmission line (as a resonator) may have only oneharmonic (one frequency resonance mode) at about 10 GHz and the nextharmonic may be about 20 GHz (which is outside the 5-15 GHz microwaveband of interest) in the state-of-the-art. Frequency resonance modesoutside the 5-15 GHz microwave band of interest are not utilized tocarry quantum information (mainly because most superconducting qubitfrequencies fall within this range (i.e., fall within the band ofinterest), and many microwave generators, measuring devices, andmicrowave components are commercially available in this range), andtherefore, multimode resonator_a 115A and multimode resonator_b 115B inembodiments may each have several tens or hundreds of frequencyresonance modes (i.e., high density of modes) between 5-15 GHz which canbe utilized to process quantum information using the multimode JPC 130.

In general, the density of modes of left-handed transmission lineresonators at a given angular resonance frequency ω is proportional tothe number of units cells in the resonator and inversely proportional tothe low-frequency bound ω_(IR).

It is to be noted that not all of the multiple resonance modes ofmultimode resonator_a and multimode resonator_b modes that fall within acertain band of interest, e.g., 5-15 GHz, strongly couple to the JRM atthe center, i.e., have an RF-current antinode at the location of theJRM. Hence, the resonance modes which strongly couple to the JRM are asubset (approximately half) of the available resonance modes within theband of interest. Consequently, not all of the resonance modes ofmultimode resonator_a and multimode resonator_b that fall within theband of interest can be utilized in order to perform three-wave mixing,which forms the basis for the various quantum information processingoperations enabled by this multimode device. In other words, the termmultiple modes of multimode resonator_a and multimode resonator_b usedin this disclosure mainly refers to those which strongly couple to JRMwithin the band of interest.

In one implementation, the multimode resonator_a 115A and multimoderesonator_b 115B may each have between 5 to 20 frequency resonance modesin the range 5-10 GHz which strongly couple to the JRM. In anotherimplementation, the multimode resonator_a 115A and multimode resonator_b115B may each have between 20-50 frequency resonance modes in the range5-10 GHz which strongly couple to the JRM. In yet anotherimplementation, the multimode resonator_a 115A and multimode resonator_b115B may each have 50-100 frequency resonance modes in the range 5-10GHz which strongly couple to the JRM.

Since the multimode resonator_a 115A and multimode resonator_b 115B mayeach have multiple resonance modes which strongly couple to the JRM inthe range 5-10 GHz (e.g. 5-100 frequency resonance modes), this allowsthe multimode JPC 130 to be useful in various intriguing applications inthe area of quantum information processing beyond the capabilities ofstandard dual-differential mode JPCs, such as generation of remoteentanglement between multiple qubits, generation of multiple pairs ofentangled photons, amplification of multiple microwave signals at thequantum limit, and performing unitary frequency conversion betweenmultiple propagating microwave signals at different frequencies.

FIG. 4 is an exemplary coplanar waveguide implementation of themultimode Josephson parametric converter 130 according to an embodiment.

The multimode JPC 130 includes multimode resonator_a 115A (left-handedtransmission line) comprising lumped-element inductors L_(a) (as theinductor L_(l)) and lumped-element capacitors C_(a) (as the capacitorsC_(l)). Similarly, the multimode JPC 130 includes multimode resonator_b115B (left-handed transmission line) comprising lumped-element inductorsL_(b) (as the inductor L_(l)) and lumped-element capacitors C_(b) (asthe capacitors C_(l)).

The multimode resonator_a 115A (left-handed transmission line) isconnected to the left and right nodes of the Josephson ring modulator150. The multimode resonator_a 115A connects to port_a 120A. In themultimode resonator_a 115A, the unit cell 205A includes two inductorsL_(a) connected to the capacitor C_(a). One end of the two inductorsL_(a) is connected to each other and the capacitor C_(a), while theother end of the inductors L_(a) is connected to the ground plane 405.This configuration of the unit cell 205A repeats N amount of times inthe multimode resonator_a 115A as shown in FIG. 4. It should be notedthat the use of two inductors in each unit cell is mainly for thepurpose of keeping the device symmetric with regard to connection toground. However, the use of one inductor connected to ground is alsocontemplated in one implementation.

The multimode resonator_b 115B (left-handed transmission line) isconnected to the top and bottom nodes of the Josephson ring modulator150. The multimode resonator_b 115B connects to port_b 120B. In themultimode resonator_b 115B, the unit cell 205B includes two inductorsL_(b) connected to the capacitor C_(b). One end of the two inductorsL_(b) is connected to each other and the capacitor C_(b), while theother end of the inductors L_(b) is connected to the ground plane 405.In the multimode resonator_b 115B, this configuration of the unit cell205B repeats M amount of times in the multimode resonator_b 115B asshown in FIG. 4. It should be noted that the use of two inductors ineach unit cell is mainly for the purpose of keeping the device symmetricwith regard to connection to ground. However, the use of one inductorconnected to ground is also contemplated in an implementation.

As discussed in FIG. 2, the ports_a 115A and ports_b 115B may be fedusing 180° hybrids 305A and 305B (not shown in FIG. 4). The ports_a 115Aand ports_b 115B may be coaxial cables or coplanar waveguides or may bemicrostrips or striplines with a center conductor and outside conductorseparated by a dielectric material. For ports_a 115A, the centerconductor connects to the left and right sides of the multimoderesonator_a 115A through coupling capacitors 110A, while the outsideconductor is connected to the ground plane 405. For ports_b 115B, thecenter conductor connects to the top and bottom sides of the multimoderesonator_b 115B through coupling capacitors 110B, while the outsideconductor is connected to the ground plane 405.

FIG. 5 is an exemplary semi-coplanar stripline implementation of themultimode Josephson parametric converter 130 according to an embodiment.

The multimode JPC 130 includes multimode resonator_a 115A (left-handedtransmission line) comprising inductor L_(a) (as the inductor L_(l)) andcapacitors C_(a) (as the capacitors C_(l)). Similarly, the multimode JPC130 includes multimode resonator_b 115B (left-handed transmission line)comprising inductors L_(b) (as the inductor L_(l)) and capacitors C_(b)(as the capacitors C_(l)).

The lumped-element side of the multimode resonator_a 115A is connectedto the left node of the Josephson ring modulator 150, while the rightnode is connected to the conducting plane 406. The lumped-element sideof the multimode resonator_a 115A and the conducting plane 406 connectto the 180 hybrid 305A. In the multimode resonator_a 115A, the unit cell205A includes inductor L_(a) connected to the capacitor C_(a). One endof the inductor L_(a) is connected to the capacitor C_(a), while theother end of the inductor L_(a) is connected to the conducting plane406. This configuration of the unit cell 205A repeats N amount of timesin the multimode resonator_a 115A as shown in FIG. 5. Although FIG. 5illustrates the left node connected to the lumped-element side of themultimode resonator_a 115A and the right node connected to theconducting plane 406, this configuration can be interchanged such thatthe lumped-element side of the multimode resonator_a 115A is connectedto the right node and the conducting plane 406 is connected to the leftnode.

The lumped-element side of the multimode resonator_b 115B is connectedto the top node of the Josephson ring modulator 150, while the bottomnode is connected to the conducting plane 407. The lumped-element sideof the multimode resonator_b 115B and conducting plane 407 connect tothe port_b 115B. In the multimode resonator_b 115B, the unit cell 205Bincludes inductor L_(b) connected to the capacitor C_(b). One end of theinductor L_(b) is connected to the capacitor C_(b), while the other endof the inductor L_(b) is connected to the conducting plane 407. Thisconfiguration of the unit cell 205B repeats M amount of times in themultimode resonator_b 115B as shown in FIG. 5. Although FIG. 5illustrates the top node connected to the lumped-element side of themultimode resonator_b 115B and the bottom node connected to theconducting plane 407, this configuration can be interchanged such thatthe lumped-element side of the multimode resonator_b 115B is connectedto the top node and the conducting plane 407 is connected to the bottomnode.

FIG. 6 is a method of configuring a microwave apparatus (such asmultimode JPC 130) according to an embodiment. Reference can be made toFIGS. 1-5.

At block 605, a first multimode resonator (i.e., multimode resonator_a115A) is connected to a Josephson ring modulator 150, where the firstmultimode resonator is made of a first left-handed transmission line.

At block 610, a second multimode resonator (i.e., multimode resonator_b115B) is connected to the Josephson ring modulator 150, where the secondmultimode resonator is made of a second left-handed transmission line.

The first multimode resonator (i.e., multimode resonator_a 115A)comprises a plurality of first resonance modes which strongly couple tothe JRM within a certain frequency band of interest, and the secondmultimode resonator (i.e., multimode resonator_b 115B) comprises aplurality of second resonance modes which strongly couple to the JRMwithin the same frequency band of interest.

A number of the plurality of first resonance modes in the firstmultimode resonator which strongly couple to the JRM within a certainfrequency band of interest is equal to a number of a plurality of secondresonance modes in the second multimode resonator which strongly coupleto the JRM within the same frequency band of interest. For example, thenumber of frequency resonance modes which strongly couple to the JRMwithin a certain frequency band of interest is equal in multimoderesonator_a 115A and multimode resonator_b 115B.

The number of the plurality of first resonance modes in the firstmultimode resonator which strongly couple to the JRM within a certainfrequency band of interest does not equal to the number of the pluralityof second resonance modes in the second multimode resonator whichstrongly couple to the JRM within the same frequency band of interest.For example, the multimode resonator_a 115A or the multimode resonator_b115B may have within a certain frequency band of interest more frequencyresonance modes which strongly couple to the JRM than the other.

The first multimode resonator_a 115A comprises N amount of first unitcells 205A, and the second multimode resonator_b 115B comprises M amountof second unit cells 205B. Neither N nor M is equal to zero. In oneimplementation, N equals M, and in another implementation N does notequal M.

Each of the first unit cells 205A and each of the second unit cells 205Brespectively comprises a capacitor (C_(a), C_(b)) connected to one endof an inductor (L_(a), L_(b)), while another end of the inductor (L_(a),L_(b)) is connected to ground 405 or conducting planes 406, 407, asdepicted is FIGS. 2, 4 and 5.

The Josephson ring modulator 150 comprises a first pair of nodes (e.g.,left and right nodes JRM 150) opposite one another and a second pair ofnodes (e.g., top and bottom nodes of the JRM 150) opposite one another.The first multimode resonator_a 115A is connected to the first pair ofnodes. One of the first pair of nodes is connected to the lumped-elementside of the resonator, and the conducting plane 406 is connected toanother one of the first pair of nodes, as depicted in FIG. 5. Thesecond multimode resonator_b 115B is connected to the second pair ofnodes. One of the second pair of nodes is connected to thelumped-element side of the resonator and the conducting plane 407 isconnected to another one of the second pair of nodes, as depicted inFIG. 5.

Each of the first unit cells 205A and each of the second unit cells 205Brespectively comprises a first inductor (first L_(a), first L_(b)), asecond inductor (second L_(a), second L_(b)), and a capacitor (C_(a),C_(b)), as depicted in FIG. 4. The first ends of the first inductor andthe second inductor are connected together, while second ends of thefirst inductor and the second inductor are connected to ground, and thecapacitor is connected to the first ends, as depicted in FIG. 4.

The Josephson ring modulator 150 comprises a first pair of nodesopposite one another and a second pair of nodes opposite one another ina Wheatstone bridge. The first multimode resonator_a 115A is connectedto the first pair of nodes, and the second multimode resonator_b 115B isconnected to the second pair of nodes.

The first unit cells are connected to one another in series, and thesecond unit cells are connected to one another in series.

In one implementation, a capacitance and an inductance in each of thefirst unit cells 205 (in the multimode resonator_a 115A) are differentfrom a capacitance and an inductance in each of the second unit cells(in the multimode resonator_b 115B). Since the unit cells 205A in themultimode resonator_a 115A are different from the unit cells 205B in themultimode resonator_b 115B, the multimode resonator_a 115A has differentresonance modes and resonance frequencies than the multimode resonator_b115B.

In another implementation, the capacitance and the inductance in each ofthe first unit cells (in the multimode resonator_a 115A) match thecapacitance and the inductance in each of the second unit cells (in themultimode resonator_b 115B).

The lumped-element inductances and capacitances used in each multimoderesonator can vary from one unit cell to another. Such perturbation tothe periodic structure of the multimode resonator can be used in orderto alter the frequency spacing between certain eigenmodes of themultimode resonator.

The lumped-element inductances used in the design of the left-handedtransmission lines of the multimode resonators, e.g. L_(a) and L_(b),can be implemented using narrow superconducting wires in a meanderconfiguration. The total inductance of the superconducting wire may be acombination of geometric and kinetic inductances. The lumped-elementinductances used in the design of the left-handed transmission lines ofthe multimode resonators can also be implemented as an array of largeJosephson Junctions.

The lumped-element capacitances used in the design of the left-handedtransmission lines of the multimode resonators, e.g. C_(a) and C_(b),can be implemented as interdigitated capacitors or plate capacitors witha dielectric layer deposited between two electrodes along the centerconductor of the left-handed transmission line.

It will be noted that various microelectronic device fabrication methodsmay be utilized to fabricate the components/elements discussed herein asunderstood by one skilled in the art. In superconducting andsemiconductor device fabrication, the various processing steps fall intofour general categories: deposition, removal, patterning, andmodification of electrical properties.

Deposition is any process that grows, coats, or otherwise transfers amaterial onto the wafer. Available technologies include physical vapordeposition (PVD), chemical vapor deposition (CVD), electrochemicaldeposition (ECD), molecular beam epitaxy (MBE) and more recently, atomiclayer deposition (ALD) among others.

Removal is any process that removes material from the wafer: examplesinclude etch processes (either wet or dry), and chemical-mechanicalplanarization (CMP), etc.

Patterning is the shaping or altering of deposited materials, and isgenerally referred to as lithography. For example, in conventionallithography, the wafer is coated with a chemical called a photoresist;then, a machine called a stepper focuses, aligns, and moves a mask,exposing select portions of the wafer below to short wavelength light;the exposed regions are washed away by a developer solution. Afteretching or other processing, the remaining photoresist is removed.Patterning also includes electron-beam lithography.

Modification of electrical properties may include doping, such as dopingtransistor sources and drains, generally by diffusion and/or by ionimplantation. These doping processes are followed by furnace annealingor by rapid thermal annealing (RTA). Annealing serves to activate theimplanted dopants.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

What is claimed is:
 1. A method of configuring a microwave apparatus,the method comprising: connecting a first multimode resonator to aJosephson ring modulator, wherein the first multimode resonator is madeof a first left-handed transmission line; and connecting a secondmultimode resonator to the Josephson ring modulator, wherein the secondmultimode resonator is made of a second left-handed transmission line.2. The method of claim 1, wherein the first multimode resonatorcomprises a plurality of first resonance modes that couple to theJosephson ring modulator within a certain frequency band; and whereinthe second multimode resonator comprises a plurality of second resonancemodes that couple to the Josephson ring modulator within the certainfrequency band.
 3. The method of claim 1, wherein a number of aplurality of first resonance modes in the first multimode resonatorwhich couple to the Josephson ring modulator within a certain frequencyband is equal to a number of a plurality of second resonance modes inthe second multimode resonator which couple to the Josephson ringmodulator within the certain frequency band; or wherein the number ofthe plurality of first resonance modes in the first multimode resonatorwhich couple to the Josephson ring modulator within a certain frequencyband does not equal to the number of the plurality of second resonancemodes in the second multimode resonator which couple to the Josephsonring modulator within the certain frequency band.
 4. The method of claim1, wherein the first multimode resonator comprises N amount of firstunit cells; wherein the second multimode resonator comprises M amount ofsecond unit cells; and wherein neither N nor M is equal to zero.
 5. Themethod of claim 4, wherein each of the first unit cells and each of thesecond unit cells comprises a capacitor connected to one end of aninductor, while another end of the inductor is connected to ground planeor conducting plane.
 6. The method of claim 5, wherein the Josephsonring modulator comprises a first pair of nodes opposite one another anda second pair of nodes opposite one another; wherein a lumped-elementside of the first multimode resonator is connected to one of the firstpair of nodes and a conducting plane is connected to another one of thefirst pair of nodes; and wherein a lumped-element side of the secondmultimode resonator is connected to one of the second pair of nodes anda conducting plane is connected to another one of the second pair ofnodes.
 7. The method of claim 4, wherein each of the first unit cellsand each of the second unit cells comprises a first inductor, a secondinductor, and a capacitor; wherein first ends of the first inductor andthe second inductor are connected together, while second ends of thefirst inductor and the second inductor are connected to ground; andwherein the capacitor is connected to the first ends.
 8. The method ofclaim 7, wherein the Josephson ring modulator comprises a first pair ofnodes opposite one another and a second pair of nodes opposite oneanother; wherein the first multimode resonator is connected to the firstpair of nodes; and wherein the second multimode resonator is connectedto the second pair of nodes.
 9. The method of claim 4, wherein either Nequals M or N does not equal M; wherein the first unit cells areconnected to one another in series; and wherein the second unit cellsare connected to one another in series.
 10. The method of claim 4,wherein a capacitance and an inductance in each of the first unit cellsare different from a capacitance and an inductance in each of the secondunit cells; or wherein the capacitance and the inductance in each of thefirst unit cells match the capacitance and the inductance in each of thesecond unit cells.